Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable Weights

and Applied Analysis 3 Lemma 2.3. Besides the properties of the argument functions, the Hamiltonian structure of the problem is essential for us to obtain variational periodic and antiperiodic halfeigenvalues, see Lemma 3.2. After introducing the rotation number function, we can easily obtain the ordering of these variational periodic half-eigenvalues, see 3.38 . For regular self-adjoint linear Sturm-Liouville problems, the continuous dependence of eigenvalues on weights or potentials in the usual L topology is well understood, and so is the Fréchet differentiable dependence. Many of these results are summarized in 7 . However, since the space of potentials or weights is infinite-dimensional, such a continuity result cannot answer many basic questions. For example, if potentials or weights are confined to a bounded set or a noncompact set, are the eigenvalues finite? To answer such kind of questions, a stronger continuity result is obtained in 8 for Sturm-Liouville operators and Hill’s operators. That is, the eigenvalues are continuous in potentials in weak topology wγ . Based on such a stronger continuity and the differentiability, variational method and singular integrals are applied in 9 to obtain the extremal value of smallest eigenvalues of Hill’s operators with potentials confined to L1 balls. The continuity result in weak topology are generalized to scalar p-Laplacian for eigenvalues on potentials see 10 , for separated eigenvalues on indefinite weights see 1 , and for half-eigenvalues on potentials see 11 . Some elementary applications are also presented in 1, 10 . In this paper, we will prove that the variational periodic or antiperiodic halfeigenvalues λ m and λ R m defined by 3.22 and 3.25 , resp. , and all the half-eigenvalues in ΣD and ΣN , are continuous in weights a, b ∈ Lγ , wγ . See Theorems 3.12, 4.3, and 5.3. Moreover, the Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in weights a, b ∈ Lγ , ‖ · ‖γ , see Theorems 4.4 and 5.3. Due to the socalled parametric resonance 12 or the so-called coexistence of periodic and antiperiodic eigenvalues 13 , periodic and antiperiodic half-eigenvalues are, in general, not differentiable in weights a, b . If λ is a half-eigenvalue of 1.1 corresponding to weights a, b and satisfying the boundary condition 1.2 , 1.3 , 1.4 , or 1.5 , then −λ is also a half-eigenvalue of 1.1 corresponding to weights −a,−b and satisfying the same boundary condition. So we need only consider nonnegative half-eigenvalues of 1.1 . Some preliminary results are given in Section 2. However, Sections 3, 4, and 5 are devoted to ΣA/P , ΣD, and ΣN , respectively. 2. Preliminary Results Given p ∈ 1,∞ , denote by Cp θ , Sp θ the unique solution of the initial value problem dx dθ −φp∗ ( y ) , dy dθ φp x , ( x 0 , y 0 ) 1, 0 . 2.1 The functions Cp θ and Sp θ are the so-called p-cosine and p-sine, because they possess some properties similar to those of cosine and sine functions, such as i both Cp θ and Sp θ are 2πp-periodic, where πp 2π p − 1 1/p p sin ( π/p ) ; 2.2 4 Abstract and Applied Analysis ii Cp θ 0 if and only if θ πp/2 mπp,m ∈ Z, and Sp θ 0 if and only if θ mπp, m ∈ Z; iii one has ∣ ∣Cp θ ∣ ∣p ( p − 1∣Sp θ ∣ ∣p ≡ 1. 2.3 Given a, b ∈ Lγ , γ ∈ 1,∞ , let y −φp ( x′ ) . 2.4 In the p-polar coordinates x rCp θ , y r2/p ∗ Sp θ , 2.5 the scalar equation ( φp ( x′ ))′ a t φp x − b t φp x− 0, a.e. t ∈ 0, 1 , 2.6 is transformed into the following equations for r and θ θ′ A t, θ;a, b : ⎧ ⎨ ⎩ a t ∣Cp θ ∣p ( p − 1∣Sp θ ∣p∗ , if Cp θ ≥ 0, b t ∣Cp θ ∣p ( p − 1∣Sp θ ∣p∗ , if Cp θ < 0, 2.7 ( log r )′ G t, θ;a, b : ⎧ ⎪⎨ ⎪⎩ p 2 a t − 1 φp ( Cp θ ) φp∗ ( Sp θ ) , if Cp θ ≥ 0, p 2 b t − 1 φp ( Cp θ ) φp∗ ( Sp θ ) , if Cp θ < 0. 2.8 For any θ0 ∈ R, denote by θ t;θ0, a, b , r t;θ0, a, b , t ∈ 0, 1 , the unique solution of 2.7 2.8 satisfying θ 0;θ0, a, b θ0 and r 0;θ0, a, b 1. Let Θ θ0, a, b : θ 1;θ0, a, b , R θ0, a, b : r 1;θ0, a, b . 2.9 As A t, θ;a, b is independent of r and is 2πp-periodic in θ, we have θ ( t;θ0 2mπp, a, b ) ≡ θ t;θ0, a, b 2mπp 2.10 for all t ∈ 0, 1 , θ0 ∈ R, and m ∈ Z. An important property of the argument solution θ is the quasimonotonicity as in the following lemma. Abstract and Applied Analysis 5 Lemma 2.1 see 14 . Let θ t θ t;θ0, a, b be a solution of 2.7 . Then θ t ≥ −p 2 mπp at t ∈ 0, 1 ⇒ θ s > − πp 2 mπp ∀s ∈ t, 1 . 2.11and Applied Analysis 5 Lemma 2.1 see 14 . Let θ t θ t;θ0, a, b be a solution of 2.7 . Then θ t ≥ −p 2 mπp at t ∈ 0, 1 ⇒ θ s > − πp 2 mπp ∀s ∈ t, 1 . 2.11 Denote by C0 : C 0, 1 ,R the space of continuous functions from 0, 1 to R. Some dependence results of solutions r and θ on a, b are collected in the following theorem. Theorem 2.2 see 11 . i The functional R × Lγ , wγ )2 −→ R, θ, a, b −→ Θ θ, a, b 2.12 is continuous. Herewγ denotes the weak topology in Lγ . ii The functional R × ( Lγ , ‖·‖γ )2 −→ R, θ, a, b −→ Θ θ, a, b 2.13 is continuously differentiable. The derivatives of Θ θ, a, b at θ, at a ∈ Lγ , and at b ∈ Lγ (in the Fréchet sense), denoted, respectively, by ∂θΘ, ∂aΘ, and ∂bΘ, are ∂θΘ θ, a, b 1 R2 θ, a, b , 2.14 ∂aΘ θ, a, b X p ∈ C0 ⊂ ( Lγ , ‖·‖γ )∗ , 2.15 ∂bΘ θ, a, b X p − ∈ C0 ⊂ ( Lγ , ‖·‖γ )∗ , 2.16